By setting it to unlimited army size, you are favouring territory and tech quality over countries that have large populations or high conscription rates. A lot of Russia's WW2 successes were based on production capability, not military quality.
Assuming Spartans retained their armed forces skill and disposition, but existed to this day using modern weapons, I think their military force would be quite formidable. Combined with your even-forces stipulation, they'd theoretically do very well in a pound-for-pound fight. However, I think France would come out on top if you were to imagine historical armies brought up to date, due to Napoleon. He may have had his flaws (being unwilling to risk his favourite elite troops, for one) but he did manage to terrorise Europe repeatedly and inspired whole coalitions against him. An invasion against Napoleon would be the most costly, as it'd probably result in a counter-invasion and an overall loss.
OK, so back to modern forces. I think ultimately it's going to boil down to the five big boys: US, Russia, UK, France and China, mainly due to their access to the nuclear option. If such an option were deployed in response to an invasion, it would instantly top all the stated categories, as it'd be as good as a loss (probably for both involved). So, out of those five...
Costliest invasion: I'm going to give this to the US- while every country has strong patriotism, I suspect the US would be the most pig-headed and unwilling to concede small defeats, combined with possibly the highest tech military overall, would result in a very costly invasion.
Bloodiest invasion: I'm inclined to go with China on this one, because of their large composition of conscripts and reserve soldiers, which would result in above average casualty numbers.
Longest Invasion: Probably the best reflection of territory; Russia has sheer size on its side, but also a rather inhospitable nature to top this category.
And the "best in show" category: Most difficult invasion: I'm British, so I'm hesitant to give it to the UK, but I'm going to for a few reasons. One I'm not entirely sure about, but given the size of the UK, I suspect our military is fairly small compared to the large countries, and as a result, our specialised units (such as special forces) will be disproportionate compared to large countries, be that SAS and other special forces, Royal Marines Commandos, or Ghurkas. Scaling up, that may give the UK a very significant advantage with your hypothetical parameters.
The UK maintains a well trained professional army by continuing to participate in armed conflicts around the world. The UK is an island nation and thus any invasion would have to be amphibious/airborne (I'd hate to think of the logistics supporting such an invasion)- combined with a balanced Army/Navy/Airforce and a small landmass, very capable of putting up heavy resistance at any point. It's also the most urbanised, urban warfare still proving to be the most difficult form of warfare to this day.